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AdvancesinAppliedMathematics
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,2022,11(4),1764-1780
PublishedOnlineApril2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.114193
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ExistenceofPositiveSolutionsto
NonlinearIntegralEquationswith
WeightsontheBoundedDomains
oftheHeisenbergGroupin
SubcriticalCase
∗
Email:jnchen@zjnu.edu.cn
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2022,11(4):1764-1780.DOI:10.12677/aam.2022.114193
•
Z
V
JianiChen
∗
CollegeofMathematicsandComputerSciencesofZhejiangNormalUniversity,JinhuaZhejiang
Received:Mar.18
th
,2022;accepted:Apr.12
th
,2022;published:Apr.20
th
,2022
Abstract
Thispaperisdevotedtoakindofnonlinearintegralequationswithweightsrelated
tothesharpHardy-Littlewood-Sobolev(hereinafterreferredtoasHLS)inequalityon
theboundeddomainsoftheHeisenberggroup
H
n
:
f
q
−
1
(
ξ
) =
Z
Ω
G
(
ξ
)
f
(
η
)
G
(
η
)
|
η
−
1
ξ
|
Q
−
α
dη
+
λ
Z
Ω
f
(
η
)
|
η
−
1
ξ
|
Q
−
α
−
β
dη,ξ
∈
¯
Ω
,
where
q>
1
,
0
<α<Q
,
0
<β<Q
−
α
,
Q
=2
n
+2
isthehomogeneousdimensionof
H
n
,
λ
∈
R
,
Ω
⊂
H
n
isasmoothboundeddomainand
G
(
ξ
)
isnonnegativecontinuousin
¯
Ω
.Inthispaper,wewillstudytheexistenceresultsofthepositivesolutionsforthe
equationinsubcriticalcase
2
Q
Q
+
α
<q<
2
.
Keywords
Heisenb ergGroup,Hardy-Littlewood-SobolevInequality,Brezis-Nirenb erg-Type
Problem,IntegralEquation,SubcriticalCase,Existence
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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g
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é
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λ
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•
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A
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/
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≤
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)
f
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k
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(
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,
0
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n
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Ω;
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ž
,
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f
(
ξ
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Z
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n
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(
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|
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|
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−
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dη,I
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Ω
f
(
ξ
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Z
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(
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−
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d
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c
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z
1
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···
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) = (
z,t
)
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×
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3
8
C
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×
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þ
D
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+
$
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{
K
(
z,t
)(
z
0
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z
+
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+
t
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+2
Im
(
z
·
z
0
))
,
Ù
¥
z,z
0
∈
C
n
,
t,t
0
∈
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±
9
z
·
z
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=
n
P
j
=1
z
j
z
0
j
.
-
z
j
=
x
j
+
iy
j
,
K
(
x
1
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2
,...,x
n
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1
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2
,...,y
n
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)
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¤
H
n
˜
‡
¢
‹
I
X
.
3ù
‡
‹
I
X
¥
,
·
‚
½
±
e•
þ
|
:
X
j
=
∂
∂x
j
+2
y
j
∂
∂t
, Y
j
=
∂
∂y
j
−
2
x
j
∂
∂t
, T
=
∂
∂t
, j
= 1
,
2
,
···
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Ø
J
y
{
X
1
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2
,
···
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n
,Y
1
,Y
2
,
···
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n
,T
}
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H
n
þ
†
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•
þ
|
˜
‡
Ä
.
d
,
H
n
þ
Y
²
F
Ý
Ú
g
ý
.
Ê
.
dŽ
f
©
O
½
Â
•
∇
H
= (
X
1
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2
,
···
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n
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1
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2
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···
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n
)
Ú
∆
H
=
n
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j
=1
(
X
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j
+
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2
j
)
.
DOI:10.12677/aam.2022.1141931768
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•
Z
V
é
?
¿
:
ξ
= (
z,t
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x
+
iy,t
)
Ú
η
= (
w,s
) = (
u
+
iv,s
),
·
‚
^
|
ξ
|
= (
|
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4
+
t
2
)
1
4
5
L
«
H
n
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à
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‰
ê
.
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A
/
,
ξ
Ú
η
ƒ
m
å
l
Œ
½
Â
•
d
(
ξ,η
)=
|
η
−
1
ξ
|
.
,
,
H
n
þ
k
˜
x
g
,
+
δ
r
(
z,t
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rz,r
2
t
)
,
∀
r>
0
,
…
H
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+
à
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‘
ê´
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= 2
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+2.
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5
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H
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°
(
HLS
Ø
ª
[8,16])
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u
0
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,
K
é
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¿
¼
ê
f
,
g
∈
L
q
α
(
H
n
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k
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H
n
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H
n
f
(
ξ
)
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(
η
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|
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−
1
ξ
|
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−
α
dηdξ
≤
D
n,α
k
f
k
L
q
α
(
H
n
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k
g
k
L
q
α
(
H
n
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¤
á
,
Ù
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D
n,α
:= (
π
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+1
2
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−
1
n
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−
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α
2
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Γ
2
(
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+
α
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)
;
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ª
¤
á
…
=
f
(
ξ
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c
1
g
(
ξ
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c
2
H
(
δ
r
(
ζ
−
1
ξ
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,
Ù
¥
c
1
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2
∈
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r>
0
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ζ
∈
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(
Ø
š
f
≡
0
½
g
≡
0
)
…
H
½
Â
•
H
(
ξ
) =
H
(
z,t
) = [(1+
|
z
|
2
)
2
+
t
2
]
−
Q
+
α
4
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·
K
1(
Ú
n
8.8[3]
Ú
Ú
n
1.3[17])
é
?
¿
ξ
∈
H
n
.
X
J
|
ξ
|≤
1
,
@
o
k
ξ
k≤|
ξ
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ξ
k
1
2
,
ù
p
k·k
´
î
A
p
‰
ê
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·
K
2(
n
Ø
ª
(
Ú
n
8.9[3]
Ú
·
K
1.4[17]))
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3
˜
‡
~
ê
C
≥
1
,
¦
é
?
¿
ξ,η
∈
H
n
Ñ
k
|
ξ
+
η
|≤
C
(
|
ξ
|
+
|
η
|
)
,
|
ξη
|≤
C
(
|
ξ
|
+
|
η
|
)(2.2)
¤
á
,
ù
p
ξ
+
η
L
«
Ê
Ï
•
þ
ƒ
\
.
·
K
3(
Ú
n
8.10[3]
Ú
·
K
1.15[17])
X
J
f
´
˜
‡
λ
g
à
g
¼
ê
,
f
∈
C
2
…
l
0
,
Ù
¥
λ
∈
R
,
@
o
•
3
˜
‡
~
ê
C>
0
,
¦
|
f
(
ξη
)
−
f
(
ξ
)
|≤
C
|
η
||
ξ
|
λ
−
1
,
ù
p
|
η
|≤
1
2
|
ξ
|
,
(2.3)
|
f
(
ξη
)+
f
(
ξη
−
1
)
−
2
f
(
ξ
)
|≤
C
|
η
|
2
|
ξ
|
λ
−
2
,
ù
p
|
η
|≤
1
2
|
ξ
|
.
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DOI:10.12677/aam.2022.1141931769
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f,g
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f
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g
(
ξ
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Z
f
(
η
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g
(
η
−
1
ξ
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dη
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f
(
ξη
−
1
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(
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½
Â
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(
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(
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−
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f
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¼
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¤
k
g
∈
C
∞
0
(
H
n
\{
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)
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k
F
(
g
) =
R
fgdξ
¤
á
.
·
K
4(
Ú
n
8.7[3])
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λ
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à
g
©
Ù
,
Ù
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0
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o
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0
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g
→
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F
Ñ
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±
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l
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p
L
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1
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loc
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,
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p
1
q
=
1
p
−
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−
1
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1
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[3])
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|
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|
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,
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,
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1
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+
∞
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1
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1
p
−
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,
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¡
(
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C
¤
k
g
∗|
ξ
|
α
−
Q
k
L
q
≤
C
k
g
k
L
p
.
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•
,
·
‚
‰
Ñ
H
n
þ
Lipschitz
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m
x
Γ
α
½
Â
9
Ù
$
^
.
½
Â
3(Lipschitz
˜
m
[3,17])
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é
u
0
<α<
1
,
Γ
α
=
{
f
∈
L
∞
∪
C
: sup
ξ,η
|
f
(
ξη
)
−
f
(
ξ
)
|
|
η
|
α
<
∞}
.
(ii)
é
u
α
= 1
,
Γ
1
=
{
f
∈
L
∞
∪
C
: sup
ξ,η
|
f
(
ξη
)+
f
(
ξη
−
1
)
−
2
f
(
ξ
)
|
|
η
|
<
∞}
.
(iii)
é
u
α
=
k
+
α
0
,
ù
p
k
´
˜
‡
ê
…
0
<α
0
≤
1
,
Γ
α
=
{
f
∈
L
∞
∪
C
:
f
∈
Γ
α
0
…
é
¤
k
D
∈
B
k
,
Ñ
k
Df
∈
Γ
α
0
}
,
Ù
¥
B
k
=
{
L
a
1
L
a
2
···
L
a
j
: 1
≤
a
i
≤
2
n, i
= 1
,
2
,
···
,j, j
≤
k
}
,
L
j
=
X
j
±
9
L
j
+
n
=
Y
j
,
j
= 1
,
2
,
···
,n.
·
K
5(
½
n
20.1[3])
é
u
0
<α<
∞
,
Γ
α
⊂
C
α
2
(
loc
)
.
DOI:10.12677/aam.2022.1141931770
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´
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f
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f
∈
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2
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Q
+
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{
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G,α,
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f
j
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{
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f
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k
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2
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m
…
¼
ê
G
(
ξ
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∈
C
(
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±
·
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k
k
G
(
ξ
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f
j
(
η
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(
η
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k
L
2
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Q
+
α
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≤
C,
…
d
f
;
5
Œ
:
•
3
{
f
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f
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(
E,
^
{
f
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L
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Ú
˜
‡
¼
ê
f
∈
L
2
Q
Q
+
α
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¦
j
→
+
∞
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,
3
L
2
Q
Q
+
α
(Ω)
¥
k
f
j
*f
,
ù
¿
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j
→
+
∞
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3
L
2
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+
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¥
k
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(
ξ
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f
j
(
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(
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(
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©
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G,α,
Ω
f
j
(
ξ
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I
1
G,α,
Ω
f
j
(
ξ
)+
I
2
G,α,
Ω
f
j
(
ξ
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:=
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f
j
(
ξ
)
}
Â
ñ
5
.
˜
•
¡
,
5
¿
|
ξ
|
α
−
Q
χ
{|
ξ
|
>ρ
}
∈
L
2
Q
Q
−
α
(Ω)
Ú
(3.1),
d
f
Â
ñ
½
Â
Œ
I
1
G,α,
Ω
f
j
(
ξ
)
Å
:
Â
ñ
I
1
G,α,
Ω
f
(
ξ
),
d
d
í
|
I
1
G,α,
Ω
f
j
(
ξ
)
|
t
Å
:
Â
ñ
|
I
1
G,α,
Ω
f
(
ξ
)
|
t
.
,
˜
•
¡
,
Ï
•
|
I
1
G,α,
Ω
f
j
(
ξ
)
|≤k
G
(
ξ
)
f
j
(
η
)
G
(
η
)
k
L
2
Q
Q
+
α
(Ω)
k|
ξ
|
α
−
Q
χ
{|
ξ
|
>ρ
}
k
L
2
Q
Q
−
α
(Ω)
≤
C
(
ρ
)
,
ù
`
²
|
I
1
G,α,
Ω
f
j
(
ξ
)
|
t
≤
C
t
(
ρ
),
ù
p
C
(
ρ
)
†
f
j
Ã
'
,
¤
±
d
Lebesgue
›
›
Â
ñ
½
n
,
Œ
Z
Ω
|
I
1
G,α,
Ω
f
j
(
ξ
)
|
t
→
Z
Ω
|
I
1
G,α,
Ω
f
(
ξ
)
|
t
,
d
d
k
I
1
G,α,
Ω
f
j
(
ξ
)
k
L
t
(Ω)
→k
I
1
G,α,
Ω
f
(
ξ
)
k
L
t
(Ω)
.
DOI:10.12677/aam.2022.1141931771
A^
ê
Æ
?
Ð
•
Z
V
Ï
d
,
·
‚
Œ
±
Ñ
3
L
t
(Ω)
¥
k
I
1
G,α,
Ω
f
j
(
ξ
)
→
I
1
G,α,
Ω
f
(
ξ
)
.
(3.2)
1
Ú
.
e
5
,
·
‚
5
©
Û
{
I
2
G,α,
Ω
f
j
(
ξ
)
}
Â
ñ
5
.
•
y
²
3
L
t
(Ω)
¥
k
I
2
G,α,
Ω
f
j
(
ξ
)
→
I
2
G,α,
Ω
f
(
ξ
),
·
‚
I
‡
y
²
j
→
+
∞
ž
k
k
I
2
G,α,
Ω
f
j
(
ξ
)
−
I
2
G,α,
Ω
f
(
ξ
)
k
L
t
(Ω)
=
k
I
2
G,α,
Ω
(
f
j
(
ξ
)
−
f
(
ξ
))
k
L
t
(Ω)
→
0
.
Š
â
Young
Ø
ª
,
·
‚
Œ
±
í
Ñ
k
I
2
G,α,
Ω
(
f
j
(
ξ
)
−
f
(
ξ
))
k
L
t
(Ω)
≤
C
k
G
(
ξ
)
G
(
η
)(
f
j
(
η
)
−
f
(
η
))
k
L
2
Q
Q
+
α
(Ω)
k|
ξ
|
α
−
Q
χ
{|
ξ
|
<ρ
}
k
L
s
(Ω)
≤
C
k
f
j
(
η
)
−
f
(
η
)
k
L
2
Q
Q
+
α
(Ω)
k|
ξ
|
α
−
Q
χ
{|
ξ
|
<ρ
}
k
L
s
(Ω)
,
(3.3)
ù
p
1
t
+1 =
1
2
Q
Q
+
α
+
1
s
,
s
= (
1
t
+
Q
−
α
2
Q
)
−
1
<
Q
Q
−
α
…
k|
ξ
|
α
−
Q
χ
{|
ξ
|
<ρ
}
k
L
s
(Ω)
≤
Cρ
β
,
Ù
¥
β
=
Q
(
1
s
−
Q
−
α
Q
).
K
(3.3)
=
z
¤
k
I
2
G,α,
Ω
(
f
j
(
ξ
)
−
f
(
ξ
))
k
L
t
(Ω)
≤
Cρ
β
.
d
ž
,
À
ρ
•
v
ê
,
@
o
j
→
+
∞
k
k
I
2
G,α,
Ω
(
f
j
(
ξ
)
−
f
(
ξ
))
k
L
t
(Ω)
→
0
,
=
j
→
+
∞
ž
,
3
L
t
(Ω)
¥
k
I
2
G,α,
Ω
f
j
(
ξ
)
→
I
2
G,α,
Ω
f
(
ξ
)
.
(3.4)
Ï
d
,
d
(3.2)
Ú
(3.4)
Œ
j
→
+
∞
ž
,
3
L
t
(Ω)
¥
k
I
G,α,
Ω
f
j
(
ξ
)
→
I
G,α,
Ω
f
(
ξ
)
.
Ú
n
y
.
2
3
þ
ã
Ú
n
1
Ä
:
þ
,
·
‚
ò
y
²
e
¡
Ú
n
,
d
d
Œ
±
í
Ñ
½
n
1
•
3
5
(
J
.
•
{
ü
å
„
,
·
‚
•y
²
λ
= 0
ž
œ
/
.
T
y
²
•{
†
[16]
¥
Ú
n
4.2
y
²
•{
ƒ
Ó
.
Ú
n
2
é
u
q>q
α
,
þ(
.
D
G,α,q
(Ω) :=sup
f
∈
L
q
(Ω)
\{
0
}
R
Ω
R
Ω
G
(
ξ
)
f
(
ξ
)
|
η
−
1
ξ
|
α
−
Q
f
(
η
)
G
(
η
)
dηdξ
k
f
k
2
L
q
(Ω)
U
L
q
(Ω)
¥
,
˜
š
K
¼
ê
ˆ
,
Ù
¥
0
<α<Q
…
¼
ê
G
(
ξ
)
∈
C
(
¯
Ω)
.
y
²
˜
m
©
,
·
‚
ò
5
y
²
D
G,α,q
(Ω)
≤
C<
+
∞
.
DOI:10.12677/aam.2022.1141931772
A^
ê
Æ
?
Ð
•
Z
V
Ï
•
G
(
ξ
)
∈
C
(
¯
Ω),
q>q
α
,
f
∈
L
q
(Ω)
…
e
f
(
ξ
) :=
(
f
(
ξ
)
ξ
∈
Ω
,
0
ξ
∈
H
n
\
Ω
,
¤
±
|
G
(
ξ
)
|≤
C
,
f
∈
L
q
α
(Ω)
…
e
f
(
ξ
)
∈
L
q
α
(
H
n
),
ù
p
k
e
f
(
ξ
)
k
L
q
α
(
H
n
)
=
k
f
(
ξ
)
k
L
q
α
(Ω)
.
é
?
¿
f
∈
L
q
(Ω),
Š
â°
(
HLS
Ø
ª
(2.1),
·
‚
k
h
I
G,α,
Ω
f,f
i
=
Z
Ω
f
(
ξ
)(
Z
Ω
G
(
ξ
)
f
(
η
)
G
(
η
)
|
η
−
1
ξ
|
Q
−
α
dη
)
dξ
≤
C
Z
Ω
Z
Ω
f
(
ξ
)
f
(
η
)
|
η
−
1
ξ
|
Q
−
α
dηdξ
=
C
Z
H
n
Z
H
n
e
f
(
ξ
)
e
f
(
η
)
|
η
−
1
ξ
|
Q
−
α
dηdξ
≤
C
·
D
n,α
k
e
f
k
2
L
q
α
(
H
n
)
=
C
·
D
n,α
k
f
k
2
L
q
α
(Ω)
≤
C
k
f
k
2
L
q
(Ω)
,
Ï
d
D
G,α,q
(Ω) :=sup
f
∈
L
q
(Ω)
\{
0
}
R
Ω
R
Ω
G
(
ξ
)
f
(
ξ
)
|
η
−
1
ξ
|
α
−
Q
f
(
η
)
G
(
η
)
dηdξ
k
f
k
2
L
q
(Ω)
=sup
f
∈
L
q
(Ω)
\{
0
}
R
Ω
f
(
ξ
)(
R
Ω
G
(
ξ
)
f
(
η
)
G
(
η
)
|
η
−
1
ξ
|
α
−
Q
dη
)
dξ
k
f
k
2
L
q
(Ω)
=sup
f
∈
L
q
(Ω)
\{
0
}
h
I
G,α,
Ω
f,f
i
k
f
k
2
L
q
(Ω)
≤
sup
f
∈
L
q
(Ω)
\{
0
}
C
k
f
k
2
L
q
(Ω)
k
f
k
2
L
q
(Ω)
=
C<
+
∞
,
=
D
G,α,q
(Ω)
≤
C<
+
∞
.
e
5
,
·
‚
‡
y
²
þ(
.
D
G,α,q
(Ω)
U
L
q
(Ω)
¥
,
˜
š
K
¼
ê
ˆ
.
3
L
q
(Ω)
¥
À
˜
‡
š
K
4
Œ
z
S
{
f
j
}
+
∞
j
=1
,
…
?
˜
Ú
¦
Ù
I
O
z
¦
k
f
j
k
L
q
(Ω)
= 1,
ù
ž
·
‚
Œ
±
lim
j
→
+
∞
Z
Ω
Z
Ω
G
(
ξ
)
f
j
(
ξ
)
|
η
−
1
ξ
|
α
−
Q
f
j
(
η
)
G
(
η
)
dηdξ
=sup
f
∈
L
q
(Ω)
\{
0
}
Z
Ω
Z
Ω
G
(
ξ
)
f
(
ξ
)
|
η
−
1
ξ
|
α
−
Q
f
(
η
)
G
(
η
)
dηdξ
=
D
G,α,q
(Ω)
…
{
f
j
}
3
L
q
(Ω)
¥
k
.
.
q
Ï
•
g
‡
Banach
˜
m
L
q
(Ω)
¥
k
.
S
´
f
O
;
,
…
d
Ú
n
1
Œ
•
:
Ž
f
I
G,α,
Ω
ä
k
;
5
,
¤
±
·
‚
Œ
±
í
ä
Ñ
•
3
{
f
j
}
˜
‡
f
S
(
ù
p
E,
^
{
f
j
}
L
«
)
Ú
f
∗
∈
L
q
(Ω),
¦
3
L
q
(Ω)
¥
k
f
j
*f
∗
±
9
3
L
q
0
(Ω)
¥
k
I
G,α,
Ω
f
j
→
I
G,α,
Ω
f
∗
,
Ù
¥
1
<q
0
<
2
Q
Q
−
α
.
Ï
d
L
q
‰
ê
f
e
Œ
ë
Y5
,
Œ
k
f
∗
k
L
q
(Ω)
≤
liminf
j
→
+
∞
k
f
j
k
L
q
(Ω)
(3.5)
DOI:10.12677/aam.2022.1141931773
A^
ê
Æ
?
Ð
•
Z
V
Ú
lim
j
→
+
∞
h
I
G,α,
Ω
f
j
,f
j
i
=
h
I
G,α,
Ω
f
∗
,f
∗
i
.
(3.6)
@
o
,
Š
â
(3.5)
Ú
(3.6),
·
‚
k
D
G,α,q
(Ω) :=lim
j
→
+
∞
R
Ω
R
Ω
G
(
ξ
)
f
j
(
ξ
)
|
η
−
1
ξ
|
α
−
Q
f
j
(
η
)
G
(
η
)
dηdξ
k
f
k
2
L
q
(Ω)
=
lim
j
→
+
∞
R
Ω
R
Ω
G
(
ξ
)
f
j
(
ξ
)
|
η
−
1
ξ
|
α
−
Q
f
j
(
η
)
G
(
η
)
dηdξ
lim
j
→
+
∞
k
f
k
2
L
q
(Ω)
=
lim
j
→
+
∞
h
I
G,α,
Ω
f
j
,f
j
i
liminf
j
→
+
∞
k
f
j
k
2
L
q
(Ω)
≤
h
I
G,α,
Ω
f
∗
,f
∗
i
k
f
∗
k
2
L
q
(Ω)
,
ù
•
Ò
´
`
,
f
∗
´
˜
‡
4
Œ
.
Ú
n
y
.
2
N
´
y
U
þ
D
G,α,q
(Ω)
4
Œ
f
(
ξ
),
3
ƒ
˜
‡
~
ê
¦
f
œ
/
e
,
÷
v
e
•
§
:
f
q
−
1
(
ξ
) =
Z
Ω
G
(
ξ
)
f
(
η
)
G
(
η
)
|
η
−
1
ξ
|
Q
−
α
dη,ξ
∈
¯
Ω
.
(3.7)
-
g
(
ξ
) =
f
q
−
1
(
ξ
)
…
q
0
=
q
q
−
1
,
K
f
(
ξ
) =
g
1
q
−
1
(
ξ
) =
g
q
0
−
1
(
ξ
),
ù
p
f
∈
L
q
(Ω).
Ï
d
,(3.7)
=
z
¤
g
(
ξ
) =
Z
Ω
G
(
ξ
)
g
q
0
−
1
(
η
)
G
(
η
)
|
η
−
1
ξ
|
Q
−
α
dη,ξ
∈
¯
Ω
,
(3.8)
Ù
¥
2
<q
0
<p
α
…
g
∈
L
q
0
(Ω).
•
¤
½
n
1
•
e
(
Ø
y
²
,
·
‚
„
I
‡
y
²
e
¡
K
5
Ú
n
.
Ú
n
3
b
g
∈
L
q
0
(Ω)
´
•
§
(3.8)
˜
‡
)
±
9
¼
ê
G
(
ξ
)
∈
C
(
¯
Ω)
.
X
J
q
0
<p
α
,
@
o
é
u
0
<α
≤
1
,
g
∈
Γ
α
(
¯
Ω)
⊂
C
α
2
(
¯
Ω)
.
y
²
e
¡
,
·
‚
ò
©
ü
‡
Ú
½
5
y
²
g
∈
Γ
α
(
¯
Ω)
⊂
C
α
2
(
¯
Ω).
1
˜
Ú
.
Ä
k
,
·
‚
5
y
²
g
∈
L
∞
(
¯
Ω)
∪
C
(
¯
Ω).
d
H¨older
Ø
ª
,
Œ
g
(
ξ
) =
Z
Ω
G
(
ξ
)
g
q
0
−
1
(
η
)
G
(
η
)
|
η
−
1
ξ
|
Q
−
α
dη
≤
C
Z
Ω
g
q
0
−
1
(
η
)
|
η
−
1
ξ
|
Q
−
α
dη
≤
C
k
g
q
0
−
1
k
L
m
(Ω)
·k|
η
−
1
ξ
|
α
−
Q
k
L
m
0
(Ω)
=
C
k
g
q
0
−
1
k
L
s
∗
q
0
−
1
(Ω)
·k|
η
−
1
ξ
|
α
−
Q
k
L
(
s
∗
q
0
−
1
)
0
(Ω)
=
C
k
g
k
q
0
−
1
L
s
∗
(Ω)
·
(
Z
Ω
|
η
−
1
ξ
|
(
α
−
Q
)
·
(
s
∗
q
0
−
1
)
0
dη
)
1
(
s
∗
q
0
−
1
)
0
≤
C
k
g
k
q
0
−
1
L
s
∗
(Ω)
,
DOI:10.12677/aam.2022.1141931774
A^
ê
Æ
?
Ð
•
Z
V
Ù
¥
m
=
s
∗
q
0
−
1
†
m
0
= (
s
∗
q
0
−
1
)
0
p
•
Ý
ê
…
s
∗
q
0
−
1
>
Q
α
>
1.
@
o
,
X
J
·
‚
Ž
‡
y
g
∈
L
∞
(
¯
Ω),
·
‚
•
I
‡
Ø
y
•
3
,
˜
~
ê
s
∗
>
0
¦
g
∈
L
s
∗
(Ω)
…
s
∗
q
0
−
1
>
Q
α
=
Œ
.
˜
g
∈
L
∞
(
¯
Ω)
y
²
,
·
‚
Ò
Œ
±
Š
â
›
›
Â
ñ
½
n
ê
þ
í
Ñ
g
∈
C
(
¯
Ω).
e
5
,
·
‚
ò
©
n
«
œ
/
5
(
½
s
∗
Š
.
œ
/
I.
e
q
0
<
Q
Q
−
α
,
·
‚
Œ
s
∗
=
q
0
,
K
s
∗
w
,
÷
v
g
∈
L
s
∗
(Ω) =
L
q
0
(Ω)
…
s
∗
q
0
−
1
=
q
0
q
0
−
1
>
Q
α
.
œ
/
II.
e
q
0
=
Q
Q
−
α
,
-
k
=[
Q
Q
−
α
] + 1
…
q
1
=(1
−
1
k
)
Q
Q
−
α
<
Q
Q
−
α
=
q
0
,
K
k
g
∈
L
q
1
(Ω).
Š
â
(3.8)
Ú
HLS
Ø
ª
é
ó
/
ª
(2.5),
·
‚
k
g
(
ξ
)
k
L
s
∗
(Ω)
=
k
Z
Ω
G
(
ξ
)
g
q
0
−
1
(
η
)
G
(
η
)
|
η
−
1
ξ
|
Q
−
α
dη
k
L
s
∗
(Ω)
≤k
C
Z
Ω
g
q
0
−
1
(
η
)
|
η
−
1
ξ
|
Q
−
α
dη
k
L
s
∗
(Ω)
=
C
k
g
q
0
−
1
∗|
ξ
|
α
−
Q
k
L
s
∗
(Ω)
≤
C
k
g
q
0
−
1
k
L
ν
(Ω)
,
ù
p
1
s
∗
=
1
ν
−
α
Q
,
Ù
¥
1
<ν<s
∗
<
+
∞
.
Ø
”
-
ν
=
q
1
q
0
−
1
>
1,
K
1
s
∗
=
q
0
−
1
q
1
−
α
Q
=
α
(
k
−
1)
Q
.
Ï
d
,
·
‚
Œ
s
∗
=
(
k
−
1)
Q
α
,
K
s
∗
w
,
÷
v
g
∈
L
s
∗
(Ω)
…
s
∗
q
0
−
1
>
Q
α
.
œ
/
III.
e
Q
Q
−
α
<q
0
<
2
Q
Q
−
α
,
d?
·
‚
ò
|
^
S
“
•{
é
~
ê
s
∗
.
Š
â
(3.8)
Ú
HLS
Ø
ª
é
ó
/
ª
(2.5),
·
‚
Œ
±
í
Ñ
k
g
(
ξ
)
k
L
s
(Ω)
=
k
Z
Ω
G
(
ξ
)
g
q
0
−
1
(
η
)
G
(
η
)
|
η
−
1
ξ
|
Q
−
α
dη
k
L
s
(Ω)
≤k
C
Z
Ω
g
q
0
−
1
(
η
)
|
η
−
1
ξ
|
Q
−
α
dη
k
L
s
(Ω)
=
C
k
g
q
0
−
1
∗|
ξ
|
α
−
Q
k
L
s
(Ω)
≤
C
k
g
q
0
−
1
k
L
µ
(Ω)
,
(3.9)
ù
p
1
s
=
1
µ
−
α
Q
,
Ù
¥
1
<µ<s<
+
∞
.
;
X
,
·
‚
ò
$
^
þ
ã
Ø
ª
(3.9)
5
Š
S
“
.
˜
m
©
,
-
µ
=
q
0
q
0
−
1
:=
µ
1
q
0
−
1
±
9
µ
2
=
s
.
Ï
•
µ
2
=
s
=(
1
µ
−
α
Q
)
−
1
…
2
Q
Q
+
α
<µ
=
1
1
−
1
q
0
<
Q
α
,
¤
±
µ
2
=
s>
2
Q
Q
−
α
>q
0
=
µ
1
.
ò
þ
ã
L
§
?
1
S
“
:
-
µ
=
µ
i
q
0
−
1
,
K
µ
i
+1
=
s
,
Ù
¥
i
=1
,
2
,
···
.
5
¿
q
0
−
2
<
2
α
Q
−
α
…
1
t
i
+1
<
Q
−
α
2
Q
.
Ï
d
,
²
L
˜
•©
Û
,
Ø
J
Ñ
t
i
+1
>
0
ž
,
k
t
i
+1
>t
i
.
¤
±
,
3
S
“
ê
õ
g
ƒ
,
•
Ò
´
`
k
0
g
ž
,
·
‚
¬
ü
«
œ
¹
:
t
k
0
q
0
−
1
<
Q
α
Ú
t
k
0
+1
q
0
−
1
≥
Q
α
.
X
J
t
k
0
+1
q
0
−
1
>
Q
α
,
·
‚
Œ
s
∗
=
t
k
0
+1
,
@
o
s
∗
w
,
÷
v
g
∈
L
s
∗
(Ω)
…
s
∗
q
0
−
1
>
Q
α
.
X
J
t
k
0
+1
q
0
−
1
=
Q
α
,
@
o
g
∈
L
t
k
0
+1
(Ω) =
L
Q
α
(
q
0
−
1)
(Ω).
-
k
= [
2
Q
Q
−
α
]+1
±
9
q
2
= (1
−
1
k
)(
q
0
−
1)
Q
α
<
t
k
0
+1
=
Q
α
(
q
0
−
1),
d
d
·
‚
Œ
±
í
ä
g
∈
L
q
2
(Ω).
Š
â
(3.8)
Ú
HLS
Ø
ª
é
ó
/
ª
(2.5),
·
‚
˜
‡
a
q
(3.9)
Ø
ª
:
k
g
(
ξ
)
k
L
s
∗
(Ω)
≤
C
k
g
q
0
−
1
k
L
ω
(Ω)
,
ù
p
1
s
∗
=
1
ω
−
α
Q
,
Ù
¥
1
<ω<s
∗
<
+
∞
.
Ø
”
-
ω
=
q
2
q
0
−
1
>
1,
K
k
1
s
∗
=
q
0
−
1
q
2
−
α
Q
=
α
(
k
−
1)
Q
.
¤
±
,
·
‚
Œ
s
∗
=
(
k
−
1)
Q
α
,
w
,
s
∗
÷
v
g
∈
L
s
∗
(Ω)
…
s
∗
q
0
−
1
>
Q
α
.
Ï
d
,
g
∈
L
∞
(
¯
Ω)
∪
C
(
¯
Ω).
1
Ú
.
Ù
g
,
·
‚
‡
y
²
g
∈
Γ
α
(
¯
Ω),
Ù
¥
0
<α
≤
1.
ù
p
,
·
‚
ò
|
^
Lipschitz
˜
m
Γ
α
½
Â
©
±
e
ü
«
œ
/
5
?
Ø
.
œ
/
i.
é
u
0
<α<
1,
·
‚
I
‡
Ø
y
sup
ξ,γ
|
g
(
ξγ
)
−
g
(
ξ
)
|
|
γ
|
α
<
∞
.
DOI:10.12677/aam.2022.1141931775
A^
ê
Æ
?
Ð
•
Z
V
d
g
∈
L
∞
(Ω)
Ú
(3.8),
Œ
|
g
(
ξγ
)
−
g
(
ξ
)
|
=
|
Z
Ω
G
(
ξγ
)
g
q
0
−
1
(
η
)
G
(
η
)
|
η
−
1
ξγ
|
Q
−
α
dη
−
Z
Ω
G
(
ξ
)
g
q
0
−
1
(
η
)
G
(
η
)
|
η
−
1
ξ
|
Q
−
α
dη
|
≤
C
k
g
k
q
0
−
1
L
∞
(Ω)
|
Z
Ω
(
G
(
ξγ
)
|
ηγ
|
α
−
Q
−
G
(
ξ
)
|
η
|
α
−
Q
)
dη
|
≤
C
k
g
k
q
0
−
1
L
∞
(Ω)
Z
Ω
|
G
(
ξγ
)
|
ηγ
|
α
−
Q
−
G
(
ξ
)
|
η
|
α
−
Q
|
dη
≤
C
k
g
k
q
0
−
1
L
∞
(Ω)
(
Z
Ω
G
(
ξγ
)
|
ηγ
|
α
−
Q
dη
+
Z
Ω
G
(
ξ
)
|
η
|
α
−
Q
dη
)
≤
C
k
g
k
q
0
−
1
L
∞
(Ω)
(
Z
Ω
|
ηγ
|
α
−
Q
dη
+
Z
Ω
|
η
|
α
−
Q
dη
)
=
C
k
g
k
q
0
−
1
L
∞
(Ω)
[
Z
Ω
(
|
ηγ
|
α
−
Q
−|
η
|
α
−
Q
)
dη
+2
Z
Ω
|
η
|
α
−
Q
dη
]
≤
C
k
g
k
q
0
−
1
L
∞
(Ω)
[
Z
Ω
||
ηγ
|
α
−
Q
−|
η
|
α
−
Q
|
dη
+2
Z
Ω
|
η
|
α
−
Q
dη
]
.
(3.10)
©
)
Z
Ω
||
ηγ
|
α
−
Q
−|
η
|
α
−
Q
|
dη
=
Z
Ω
∩{|
η
|≥
2
|
γ
|}
||
ηγ
|
α
−
Q
−|
η
|
α
−
Q
|
dη
+
Z
Ω
∩{|
η
|≤
2
|
γ
|}
||
ηγ
|
α
−
Q
−|
η
|
α
−
Q
|
dη
Ú
Z
Ω
|
η
|
α
−
Q
dη
=
Z
Ω
∩{|
η
|≥
2
|
γ
|}
|
η
|
α
−
Q
dη
+
Z
Ω
∩{|
η
|≤
2
|
γ
|}
|
η
|
α
−
Q
dη.
e
¡
,
·
‚
ò
é
§
‚
?
1
Å
˜
O
.
˜
•
¡
,
Š
â
·
K
3
¥
ª
(2.3),
·
‚
U
O
Z
Ω
∩{|
η
|≥
2
|
γ
|}
||
ηγ
|
α
−
Q
−|
η
|
α
−
Q
|
dη
≤
C
Z
Ω
∩{|
η
|≥
2
|
γ
|}
|
γ
||
η
|
α
−
Q
−
1
dη
≤
C
|
γ
|
α
.
(3.11)
Ï
•
|
η
|≤
2
|
γ
|
,
¤
±
d
·
K
2
¥
ª
(2.2)
Œ
:
•
3
˜
‡
~
ê
C
≥
1,
¦
|
ηγ
|≤
C
(
|
η
|
+
|
γ
|
)
≤
C
(2
|
γ
|
+
|
γ
|
) = 3
C
|
γ
|
.
Ï
,
·
‚
Œ
±
O
Z
Ω
∩{|
η
|≤
2
|
γ
|}
||
ηγ
|
α
−
Q
−|
η
|
α
−
Q
|
dη
≤
Z
Ω
∩{|
η
|≤
2
|
γ
|}
|
ηγ
|
α
−
Q
dη
+
Z
Ω
∩{|
η
|≤
2
|
γ
|}
|
η
|
α
−
Q
dη
≤
Z
Ω
∩{|
ηγ
|≤
3
C
|
γ
|}
|
ηγ
|
α
−
Q
d
(
ηγ
)+
Z
Ω
∩{|
η
|≤
2
|
γ
|}
|
η
|
α
−
Q
dη
≤
C
|
γ
|
α
.
(3.12)
Ï
d
,
d
(3.11)
Ú
(3.12)
Œ
í
Ñ
Z
Ω
||
ηγ
|
α
−
Q
−|
η
|
α
−
Q
|
dη
≤
C
|
γ
|
α
.
(3.13)
,
˜
•
¡
,
·
‚
Œ
±
O
Z
Ω
∩{|
η
|≥
2
|
γ
|}
|
η
|
α
−
Q
dη
≤
C
|
γ
|
α
(3.14)
DOI:10.12677/aam.2022.1141931776
A^
ê
Æ
?
Ð
•
Z
V
Ú
Z
Ω
∩{|
η
|≤
2
|
γ
|}
|
η
|
α
−
Q
dη
≤
C
|
γ
|
α
.
(3.15)
¤
±
,
Š
â
(3.14)
Ú
(3.15),
·
‚
k
Z
Ω
|
η
|
α
−
Q
dη
≤
C
|
γ
|
α
.
(3.16)
ò
(3.13)
Ú
(3.16)
“
\
(3.10)
¥
,
|
g
(
ξγ
)
−
g
(
ξ
)
|≤
C
k
g
k
q
0
−
1
L
∞
(Ω)
|
γ
|
α
,
†
é
{
`
,sup
ξ,γ
|
g
(
ξγ
)
−
g
(
ξ
)
|
|
γ
|
α
<
∞
,
ù
L
²
0
<α<
1
ž
,
g
∈
Γ
α
(Ω).
œ
/
ii.
é
u
α
= 1,
·
‚
I
‡
y
sup
ξ,γ
|
g
(
ξγ
)+
g
(
ξγ
−
1
)
−
2
g
(
ξ
)
|
|
γ
|
<
∞
.
d
g
∈
L
∞
(Ω)
Ú
(3.8),
Œ
|
g
(
ξγ
)+
g
(
ξγ
−
1
)
−
2
g
(
ξ
)
|
=
|
Z
Ω
G
(
ξγ
)
g
q
0
−
1
(
η
)
G
(
η
)
|
η
−
1
ξγ
|
Q
−
α
dη
+
Z
Ω
G
(
ξγ
−
1
)
g
q
0
−
1
(
η
)
G
(
η
)
|
η
−
1
ξγ
−
1
|
Q
−
α
dη
−
2
Z
Ω
G
(
ξ
)
g
q
0
−
1
(
η
)
G
(
η
)
|
η
−
1
ξ
|
Q
−
α
dη
|
≤
C
k
g
k
q
0
−
1
L
∞
(Ω)
|
Z
Ω
(
G
(
ξγ
)
|
η
−
1
ξγ
|
α
−
Q
+
G
(
ξγ
−
1
)
|
η
−
1
ξγ
−
1
|
α
−
Q
−
2
G
(
ξ
)
|
η
−
1
ξ
|
α
−
Q
)
dη
|
≤
C
k
g
k
q
0
−
1
L
∞
(Ω)
|
Z
Ω
(
G
(
ξγ
)
|
ηγ
|
α
−
Q
+
G
(
ξγ
−
1
)
|
ηγ
−
1
|
α
−
Q
−
2
G
(
ξ
)
|
η
|
α
−
Q
)
dη
|
≤
C
k
g
k
q
0
−
1
L
∞
(Ω)
Z
Ω
|
G
(
ξγ
)
|
ηγ
|
α
−
Q
+
G
(
ξγ
−
1
)
|
ηγ
−
1
|
α
−
Q
−
2
G
(
ξ
)
|
η
|
α
−
Q
|
dη
≤
C
k
g
k
q
0
−
1
L
∞
(Ω)
(
Z
Ω
G
(
ξγ
)
|
ηγ
|
α
−
Q
dη
+
Z
Ω
G
(
ξγ
−
1
)
|
ηγ
−
1
|
α
−
Q
dη
+2
Z
Ω
G
(
ξ
)
|
η
|
α
−
Q
dη
)
≤
C
k
g
k
q
0
−
1
L
∞
(Ω)
Z
Ω
(
|
ηγ
|
α
−
Q
+
|
ηγ
−
1
|
α
−
Q
+
|
η
|
α
−
Q
)
dη
=
C
k
g
k
q
0
−
1
L
∞
(Ω)
[
Z
Ω
(
|
ηγ
|
α
−
Q
+
|
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−
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DOI:10.12677/aam.2022.1141931777
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DOI:10.12677/aam.2022.1141931778
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